$\delta abc$ is the image of $\delta abc$ under a dilation whose center is $p$ and scale factor is $\frac{4}{3}$.

which figure correctly shows $\delta abc$ using the solid line?

choose 1 answer:

Question

$\delta abc$ is the image of $\delta abc$ under a dilation whose center is $p$ and scale factor is $\frac{4}{3}$.

which figure correctly shows $\delta abc$ using the solid line?

choose 1 answer:

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Geometric Transformations", "Dilation Scale Factor", "Center of Dilation" ], "new_concepts": [], "current_concepts": [ "Geometric Transformations", "Dilation Scale Factor", "Center of Dilation" ] } ## Analyze the properties of the dilation $$ \begin{aligned} &\text{Center of dilation: } P \text{ (located inside } \triangle ABC\text{)} \ &\text{Scale factor: } k = \frac{4}{3} > 1 \end{aligned} $$ ## Determine the position and size of the image $$ \begin{aligned} &k > 1 \implies \triangle A'B'C' \text{ is larger than } \triangle ABC \ &k > 0 \implies \text{Each image vertex } X' \text{ lies on the ray } PX \text{ extending outward from } P \ &PX' = \frac{4}{3}PX \implies \text{The image } \triangle A'B'C' \text{ completely encloses } \triangle ABC \end{aligned} $$ ## Evaluate the given option $$ \begin{aligned} &\text{In Option A, the solid red triangle } \triangle A'B'C' \text{ is larger than the dashed blue triangle } \triangle ABC \ &\text{Each vertex } A', B', C' \text{ lies on the ray from } P \text{ through } A, B, C \text{ respectively} \ &\text{Thus, Option A correctly represents the dilation} \end{aligned} $$

(A) The solid triangle $\triangle A'B'C'$ is larger than $\triangle ABC$ and expands outward from the center $P$.

(B) The solid triangle is smaller or shifted incorrectly relative to the center $P$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the properties of the dilation

Determine the position and size of the image

Evaluate the given option

Answer: